Regular Bulk Solutions in Brane-worlds with Inhomogeneous Dust and Generalized Dark Radiation

6.5 MB·English

by  A. Herrera-Aguilar

Regular Bulk Solutions in Brane-worlds with Inhomogeneous Dust and
Generalized Dark Radiation
A. Herrera-Aguilar∗
Departamento de F´ısica, Universidad Aut´onoma Metropolitana Iztapalapa,
San Rafael Atlixco 186, CP 09340, M´exico D. F., M´exico. and
Instituto de F´ısica y Matem´aticas, Universidad Michoacana de San Nicol´as de Hidalgo,
Ciudad Universitaria, CP 58040, Morelia, Michoac´an, M´exico
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1 A. M. Kuerten†
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2 CCNH, Universidade Federal do ABC 09210-580, Santo Andr´e, SP, Brazil and

n Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma
a
J de M´exico MEX–62210, Cuernavaca, Morelos, M´exico.

9
2
Rold˜ao da Rocha‡

]
c CMCC, Universidade Federal do ABC, 09210-580, Santo Andr´e, SP, Brazil and
q
- International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
r
g
[
Fromthedynamicsofabrane-worldwithmatterfieldspresentinthebulk,thebulkmetric

1
v and the black string solution near the brane are generalized, when both the dynamics of
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inhomogeneousdust/generalizeddarkradiationonthebrane-worldandinhomogeneousdark
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radiation in the bulk as well are considered — as exact dynamical collapse solutions. Based
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0 on the analysis on the inhomogeneous static exterior of a collapsing sphere of homogeneous
.
1
dark radiation on the brane, the associated black string warped horizon is studied, as well
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5 as the 5D bulk metric near the brane. Moreover, the black string and the bulk are shown to
1
: be more regular upon time evolution, for suitable values for the dark radiation parameter in
v
i the model, by analyzing the physical soft singularities.
X
r
a
PACS numbers: 04.50.Gh, 04.50.-h
∗ aha@fis.unam.mx
† [email protected]
‡ [email protected]
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I. INTRODUCTION
Brane-world models with a single extra dimension [1, 2] are decidedly a 5D phenomenological
realization of Hoˇrava-Witten supergravity solutions [3], if the moduli effects from compact extra
dimensions can be ignored (for a review, see e. g. [4]). The Hoˇrava-Witten solution [3] can be
thought of as being effectively 5D, with an extra dimension that can be large, when collated to
the fundamental scale. They provide the basis for the well-known Randall-Sundrum (RS) brane-
world models [1, 2], that comprise the mirror symmetry and a brane with tension as well, which
counterbalancestheleverageofthenegativebulkcosmologicalconstantonthebrane,encompassing
furthermore the branes self-gravity [4]. In RS brane-world scenarios, our Universe is embedded in
a 5D bulk of type AdS [2]. The formalism to be used hereon employs a general metric for the
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brane-world, instead of the Minkowski metric in the standard RS model [2].
Brane-world black holes were comprehensively studied in Randall-Sundrum like brane-world
cosmologies [5–7], where the dynamical equations on the brane are different from the general
relativity ones. In fact, the brane-world framework presents terms that handle both the effects of
thefreegravitationalfieldinthebulkandofthebraneembeddinginthebulkaswell. Theimprintof
the nonlocal gravitational field in the bulk on the brane provides a splitting into anisotropic stress,
flux, andnonlocalenergydensity, wherethislastdeterminesthetidalaccelerationoutofthebrane,
possiblyopposingtheformationofsingularities[4]. Unlikethenonlocalenergydensityandflux,the
nonlocalanisotropicstressisnotascertainedbyanyevolutionequationonthebrane. Inparticular,
isotropy of the cosmic microwave background make the existence of the FRW background be under
risk. Adiabatic density perturbations are furthermore coupled to perturbations in the bulk field,
making an open system on the brane [7].
Bulk effects in the cosmological dynamics of brane-world scenarios have been studied [8], also
in the context of thick brane-worlds [9–13], brane-worlds with variable tension [14–17] and Kalb-
Ramond fields [18], as well as in the context of more general brane-worlds and bulks [19–23].
Moreover, consequences of the gravitational collapse were proposed in the context of brane-world
scenarios in, e. g., [24–28]. In addition, dark matter was investigated already in [29] as a bulk
effect on the brane.
Black strings can be thought as being extended objects endowed with an event horizon, in low
energy string theory [30]. The bulk metric near the brane and the black string warped horizon
along the extra dimension are here reviewed, based on previous developments [4, 31]. Originally, a
Schwarzschildblackholeonthebrane-worldwasshowntobeablackstringinahigherdimensional
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spacetime, what leads to the usual astrophysical properties of black holes to be recovered in this
scenario [31]. In this prototypical context, the Kretschmann curvature invariants diverge when the
black string event horizon is approached along the axis of the black string. Several generaliza-
tions provide attempts to preclude singularities both in the bulk and on the brane-world as well.
When variable brane tension scenarios are taken into account, the brane tension can gauge the
Kretschmann scalars involved. For instance, regular bulk solutions and black strings were obtained
in Friedmann-Robertson-Walker brane-worlds under the E¨otv¨os law [32], where the singularities
related to the McVittie metric can be partially controlled as the cosmological time elapses. Indeed,
for this type of metric the 5D physical soft singularities in the bulk are alleviated as time elapses,
providing a regular 5D bulk solution, as the 5D Kretschmann invariants do not diverge. When
other metrics are taken into account, for instance the Casadio-Fabbri-Mazzacurati metric one [33],
black strings can be still emulated [34], however the related singularities in the bulk remain, re-
gardless. In order to accomplish it, effective/perturbative approaches are usually employed, where
the black string is made to evolve from the brane-world [4, 31, 35].
Brane solutions of static black hole exteriors with 5D corrections to the Schwarzschild metric
havebeenfound,forinstance,in[36–38],andfurthermoreinthecontextwherethebulksingularities
can be removed [39]. The (Schwarzchild) black string is unstable near the AdS horizon, defining
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the so called Gregory-Laflamme instability [40, 41]. This scenario might be drastically altered
by the inhomogeneous dust and the dark radiation. In order to accomplish this effect, we use
a procedure to calculate both the metric near the brane and the 5D black string horizon [15],
uniquely from a brane-world black hole metric and the associated Weyl tensor. Based on the
knowledge of both the Sasaki-Shiromizu-Maeda effective field equations on the brane and upon the
5D Einstein and Bianchi equations [4, 7, 35, 42, 43], both the bulk metric near the brane and in
particular the black string warped horizon can be designed, by using a Taylor expansion along the
extra dimension. Such procedure provides information about all the bulk metric components [15].
Indeed, the bulk spacetime may be either given, by solving the full 5D equations or alternatively
obtainedbyevolvingthebrane-worldblackholemetricoffthebrane,whatencompassestheimprint
from the bulk via the Weyl tensor. Numerical methods have been employed to find black hole
solutions in the context of black strings and fluid/gravity correspondence [41]. Similar methods
involving expansions of the metric have been used in the context of black strings [44], disposing
the black string metric as the leading order solution in a Taylor expansion.
The bulk shape of the black string horizon has been merely investigated in very particular
cases [4, 35], and latterly the standard black string was studied in the context of a brane-world
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with variable tension [15]. Moreover, realistic models that take into account a post-Newtonian pa-
rameter on the Casadio-Fabbri-Mazzacurati black string [45], and the black string in a Friedmann-
Robertson-Walker E¨otv¨os brane-world [34], also represent interesting applications.
Recently regular black strings solutions associated to a dynamical brane-world have been ob-
tainedinthecontextofavariablebranetension[32]. Theanalysisofthe5DKretschmanninvariants
makes us capable of attenuating the bulk physical singularities along some eras of the evolution
of the Universe for the McVittie metric on an E¨otv¨os fluid brane-world. This paper is devoted to
encompass a framework with dark radiation and inhomogeneous dust. The 5D physical singulari-
ties in the bulk are shown to be inherited from the 4D brane-world and no additional singularity
appears in the bulk, for some range of parameters in our model. Nevertheless, the bulk physical
soft singularities can be unexpectedly controlled in the bulk upon time evolution, what makes a
regular bulk 5D solution in most ranges of the dark radiation parameter.
This paper is organized as follows: in Section II the dark radiation dynamics on the brane is
analyzed and reviewed. Starting with the Lemaˆıtre-Tolman-Bondi (LTB) metric on the brane,
the effective field equations for dark radiation on the brane are solved. The dynamical radiation
model is shown to mimic a 4D cosmological constant on the brane. Both the black string solution
and the bulk metric are obtained thereon. After obtaining the standard dark radiation model,
a generalized framework is proposed. Both associated metrics are derived. In Section III the
5D bulk metric near the brane and the generalized black string are derived and studied, and in
Section IV the black string warped horizon in the context of inhomogeneous dust and generalized
dark radiation is studied. Moreover, the black string physical singularities are analyzed from the
Kretschmann invariants. The 5D physical singularities in the bulk reflect the 4D brane-world
physical singularities. We analyze further the Kretschmann scalars generated by higher order
derivatives of the Riemann tensor, and the respective physical soft singularities show that the bulk
5D solution is regular, in some ranges of the dark radiation parameter.
In order to fix the notation hereupon µ,ν = 0,1,2,3 and M,N = 0,1,2,3,5, and let n be a
time-like covector field normal to the brane and y the associated Gaussian coordinate. The brane
metric components g and the corresponding components of the bulk metric gˇ are related by
µν µν
g +n n = gˇ [4]. With these choices we can write g = 1 and g = 0, and thus the 5D bulk
µν µ ν µν 55 µ5
metric reads
gˇ dxM dxN = g (xα,y)dxµdxν +dy2, (1)
MN µν
where M,N = 0,1,2,3 effectively.
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The initial paradigm concerning a perturbative method for obtaining the black string solution
consisting in assuming the Schwarzschild form for the induced brane metric, on a RS brane-world.
Subsequently a sheaf of such solutions are disposed into the extra dimension [31]:
(cid:32) (cid:33)
(cid:18) 2GM(cid:19) dr2
(5)ds2 = e−2|y|/(cid:96) − 1− dt2+ +r2dΩ +dy2 , (2)
r 1− 2GM 2
r
(cid:113)
where (cid:96) = −6 denotes the curvature radius of the bulk AdS , wherein the RS brane-world is
Λ5 5
embedded. Each space of constant y is a 4D Schwarzschild spacetime with a singularity along
r = 0 for all y, the well-known (Schwarzschild) black string.
AsitisgoingtobeclearerinSectionIII,thearealradiusofthesheafofsuchsolutionsalongthe
extra dimension is called the black string warp horizon, that shall be precisely defined in Section
III.
II. DARK RADIATION DYNAMICS ON THE BRANE
Henceforward some results concerning the dark radiation dynamics on the brane will be shortly
revisited [46–48], in order to briefly introduce the framework to get both the bulk metric near the
brane and the black string encompassing the dark radiation parameter and the effective cosmolog-
ical constant. New black strings solutions are here derived in the scenario provided in [46–48] on a
background described by a Lemaˆıtre-Tolman-Bondi (LTB) metric [49–51]. A solution for the black
string that has as limit a tidal Reissner-Nordstr¨om black hole solution on the brane was obtained
in [46], in the Randall-Sundrum scenario. In order to work with the effective Einstein equations on
the brane, some conditions on the projected Weyl tensor are usually assumed, in order to provide
a closed system. Besides, by taking into account a system of equations where a specific state equa-
tion leads to a inhomogeneous density that has precisely the dark radiation form [52–54] (and its
generalizations for thick branes [13, 55]), and by solving the effective Einstein equations, the LTB
metric can be derived. Subsequently both the bulk near the brane and the black string solution is
obtained are obtained in this context.
From the dynamics of a brane-world with matter fields present in the bulk [47], the associated
blackstringsolutionwillbeshowntopresentageneralizeddarkradiationform. Thewaytoobtain
the LTB metric is essentially different from that acquired in [46]. The inhomogeneous density is
associated with conformal bulk fields, instead of being related to the electric part of the Weyl
tensor. The black string solution is similarly obtained by a change in the coordinate system and
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its final form generalizes the first case. Thereafter two new black string solutions will be presented
and subsequently used in the construction of the horizon profile in the bulk.
A. The Lemaˆıtre-Tolman-Bondi Metric
The Lemaˆıtre-Tolman-Bondi (LTB) metrics are exact solutions of the Einstein equations that
describe inhomogeneous spacetimes, having dust as the source. This type of models consider
inhomogeneous generalizations of the Friedmann-Robertson-Walker (FRW) metrics, as the LTB
metrics. An alternative approach for the LTB space-time is based in evolution equations of co-
variant objects, as the density, expansion scalar, electric Weyl tensor, shear tensor and spatial
curvature. The dynamics is reduced to scalar equations, and the FRW spacetime is achieved when
two of these scalars associated with the shear tensor and electric Weyl tensor are zero. This formu-
lation is based on a 1+3 covariant description [56], which can be further applied to the LTB model
[57, 58]. In general, the applications of these models involve black holes, galaxy clusters, super-
clusters, cosmic voids, supernovas and redshift drift, for instance [59]. Initially found by Lemaˆıtre
[49], the LTB metric describes a spherically symmetric inhomogeneous fluid with anisotropic pres-
surecosmological constant are present for instance in the Tolman model [50]. To derive the LTB
solution, in comoving coordinates the general form for the line element is given by:
ds2 = g dxµdxν = −dt2+A2(r,t)dr2+R2(r,t)dΩ , (3)
(4) µν 2
where the 2-sphere area element is denoted by dΩ , the energy-momentum tensor written as T µ =
2 ν
diag(−ρ,−ρ ,−ρ ,−ρ ), where ρ denotes the energy density, Λ denotes the brane cosmological
Λ Λ Λ 4
constant with associated energy density ρ = κ−2Λ , and κ is the 4D gravitational coupling
Λ 4 4 4
constant. The Einstein equations, for each one of the space diagonal components, are given by the
following expressions:
(cid:40) (cid:34) (cid:35)(cid:41)
1 1 (cid:18)∂ R(cid:19)2
2∂2R+ 1+(∂ R)2− r = Λ , (4)
R t R t A 4
∂2R ∂ R∂ A ∂ R∂ A ∂2A ∂2R
t + t t + r r + t − r ∂ ∂ R = Λ , (5)
R R A R A3 A RA2 t r 4
∂ A
t
∂ R = Λ . (6)
r 4
A
The function A(r,t) = g(r)∂ R satisfies Eq.(5). By setting g(r) = (1+f(r))−1/2, the usual form
r
of the LTB metric is hence obtained:
(∂ R)2
ds2 = −dt2+ r dr2+R2dΩ , (7)
(4) 1+f 2
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where f(r) > −1. The function f can be interpreted as the energy density shell f(r) = 2E(r). The
function g(r) is a geometric factor such that when g(r) = 1 the spatial sections are flat. Eqs.(4)
and (5) are not independent, leading to the expression
2M Λ
(∂ R)2 = f + + 4R2, (8)
t
R 3
where M = M(r) is an arbitrary function of integration that gives the gravitational mass within
each comoving shell of coordinate radius r. By definition of mass in [51], one can write 2dm/dr =
√
k2ρAR2, which implies M = (cid:82) d m 1+fdr. The non-relativistic limit f (cid:28) 1 yields M ∼ m, and
4 r
Eq.(8) reads 1 (∂ R)2− M = f. The first term is interpreted as kinetic energy, the second stands
2 t R 2
for the Newtonian potential term and f is twice the energy of the system, when Λ = 0. Hence
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M is the relativistic generalization of the Newtonian mass. When f is negligible, namely in the
non-relativistic limit, the spatial sections are flat when g = 1. The function g provides the energy
in each spatial section, and thus carries the information of curvature for each section.
Finally, Eq. (8) implies that
(cid:90) R(cid:18) 2M Λ R2(cid:19)−1/2
4
t−t (r) = f + + dR, (9)
N
R 3
0
where t is known as the “bang time”. Eq. (8) can be used to classify the LTB models into three
N
classes. When Λ = 0 it reads:
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−1 < f < 0 elliptic,
f = 0 parabolic,
f > 0 hyperbolic.
When Λ (cid:54)= 0 the potential V(R) = 2M + Λ4R2 leads to a different classification, depending on
4 R 3
the sign of Λ .
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B. LTB Solution on the Brane
In this Subsection the LTB solution associated with dark radiation on the brane is reviewed,
starting with the projected Einstein equations on the brane in vacuum. Unlike the Reissner-
Nordstr¨om black hole, this new solution has a specific dark radiation tidal charge Q. The 4D and
5D coupling constants are related by κ2 = 1λκ4. The field equations in 5D Einstein theory lead
4 6 5
to the projected equations [42, 60]
G = −Λ g +κ2T +κ4Π −E , (10)
µν 4 µν 4 µν 5 µν µν
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where Π is a term quadratic in the energy-momentum T and provides high-energy corrections
µν µν
arising from the extrinsic curvature of the brane, what increases the pressure and effective density
of collapsing matter. The term E is the projection of the bulk Weyl tensor and provide Kaluza-
µν
Klein corrections originated from 5D graviton stresses [28], as the massive modes for the graviton
in the linearized regime. For observers on the brane such stresses are nonlocal, in the sense
that they are local density inhomogeneities on the brane generate Weyl curvature in the bulk
backreacting nonlocally on the brane [4, 7, 16, 17, 53, 54]. Therefore the vacuum equations are
G = −Λ g −E , where the Weyl projected tensor can be identified with a trace-free energy-
µν 4 µν µν
momentum as E ∼ κ2T , provided by [61]
µν 4 µν
(cid:18)κ (cid:19)4(cid:20) (cid:18) 1 (cid:19) (cid:21)
5
E = U v v + h +P +2Q v , (11)
µν κ µ ν 3 µν µν (µ ν)
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where U is the effective energy, P is an anisotropic stress tensor, Q is the effective energy flux,
µν µ
v isa4Dvelocityvectorsatisfyingv vµ = −1, andh issuchthatvµh = 0, beingthuspossible
µ µ µν µν
to write h = g +v v . A non-static spherically symmetric brane-world with P (cid:54)= 0 can be
µν µν µ ν µν
described by the line element
ds2 = −C2(r,t)dt2+B2(r,t)dr2+R2(r,t)dΩ (12)
(4) 2
with Q = 0. The anisotropic stress tensor P can be represented by P = P (cid:0)r r + 1h (cid:1),
µ µν µν µ ν 3 µν
where P = P(r,t) is a scalar field and r is the unit radial vector. With these assumptions the
µ
electric part of the Weyl tensor yields
(cid:18)κ (cid:19)4
E ν = 5 diag(ρ,−p ,−p ,−p ), (13)
µ κ r T T
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with ρ = U, 3p = U +2P and 3p = U −P.
r T
By assuming the brane field equation ∇µE = 0 [4] and by considering the state equation
µν
ρ = −p , it yields ∂ U +4∂tRU = 0 = ∂ U +4∂rRU [46], implying that
r t R r R
(cid:18)κ (cid:19)4 Q
5
U = , (14)
κ R4
4
where the constant Q is the dark radiation tidal charge [46–48]. As ρ = U, the energy density in
this case is related to the inhomogeneous density. Thus the 4D Einstein equations (10) read
Q
G = −Λ g − (v v −2r r +h ). (15)
µν 4 µν R4 µ ν µ ν µν
It follows, by solving the above equations, that the component G is obtained by the expression
tr
∂ B∂ R−B∂ ∂ R = 0, and therefore the function
t r t r
B = H−1∂ R (16)
r
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satisfies this relation with H = H(r). Considering thus such expression for B in the trace equation
−Gt+Gr +2Gθ [46] it is possible to write the expression as follows
t r θ
Q Λ
(∂ R)2 = f − + 4R2, (17)
t R2 3
which is similar to Eq.(8). Integrating Eq. (17), it reads
(cid:90) (cid:18) Q Λ R2(cid:19)−1/2
4
±t+τ(r) = f − + dR, (18)
R2 3
which is analogous to Eq. (9). It is thus possible to write (12) in the LTB form given by (7).
Making the transformation of the LTB coordinates (t,r) to curvature coordinates (T,R) as
(cid:113)
(cid:90) R Λ4 −Q
3
T = t+ dR, (19)
Λ4R4−R2−Q
3
the following 4D metric is finally obtained:
(cid:18) Q Λ (cid:19) (cid:18) Q Λ (cid:19)−1
ds2 = − 1+ − 4R2 dT2+ 1+ − 4R2 dR+R2dΩ . (20)
(4) R2 3 R2 3 2
This metric is known as the inhomogeneous static exterior of a collapsing sphere of homogeneous
dark radiation [28, 62]. Note that when Λ = 0, the solution (20) is formally analogous to the
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Reissner-Nordstr¨om solution, when one identifies the electric charge to the dark radiation tidal
charge.
In what follows this solution will be generalized, by considering a generalized dark radiation
term with dark radiation charge Q where η is a parameter characterizing the model of the dark
η
radiation[47]. Thedynamicsofasphericallysymmetricbrane-worldisalsoanalyzed,whenthebulk
a)carriesmatterfields; andb)whenitswarpfactorcharacterizesaglobalconformaltransformation
consistent with Z symmetry. Finally, it is possible to study the bulk metric and the black string
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solutionswithatermanalogoustotheblackholesolutionwithcosmologicalconstantonthebrane.
In this framework, the energy-momentum tensor encompasses conformal bulk matter fields, whose
dynamics provide a specific state equation [46–48].
Considernowageneralconformalsphericallysymmetricmetricd˚s2 consistentwithZ symmetry
5 2
along the extra dimension on the brane
d˚s2 = Ω2(cid:0)−e2Adt2+e2Bdr2+R2dΩ +dz2(cid:1), (21)
5 2
where z stands for the conformal extra dimensional coordinate, and A,B,R, and Ω are general
functions of the coordinates (t,r,z). Ω denotes the conformal factor. The Einstein field equations
are given by
(cid:104) (cid:105)
G˚ = −κ2 −T˚ +λδ(z−z )˚g +Λ δ , (22)
MN 5 MN 0 MN 5 MN
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where˚g denotesthecomponentsoftheistheinducedmetricandT˚ standsforthecomponents
MN µν
of the energy-momentum tensor representing the bulk fields. The brane is localized at z = z .
0
UndertheconformaltransformationT˚MN = ΩsTMN, theenergy-momentumtensorisassumed,
as usual, to have weight s = −4 [46]. The conformal Einstein tensor is given by
G˚ = G +Υ (gˇ). (23)
MN MN MN
By using the expression˚gM = Ω−1gM, Eq.(22) hence leads to the following equations:
N N
GM = κ2T M, (24)
N 5 N
ΥM = −κ2Ω2(cid:2)Ω−1λδ(z−z )gM +Λ δM(cid:3). (25)
N 5 0 N 5 N
Eq.(24) evinces that the 5D Einstein tensor is related solely to the presence of fields in the bulk,
and is independent both on the brane tension λ and upon the bulk cosmological constant Λ .
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Moreover, Eq. (25) provides the dynamics of the conformal factor. The divergence condition leads
to ∇ TM +Ω−1(cid:2)3TM∂ Ω−T∂ Ω(cid:3) = 0, and from the Bianchi identity Eqs. (24) and (25) it
M N N M N
implies that
either ∇ TM = 0 (26)
M N
or 3T M∂ Ω = T∂ Ω. (27)
N M N
Hence Eqs. (26) and (27) require that 2T z = T µ. Now, by considering the energy-momentum
z µ
tensor
T M = diag(−ρ,p ,p ,p ,p ) , (28)
N r T T z
the state equation ρ−p −2p +2p = 0 holds. It implies that ∇ T z = 0 and subsequently that
r T z z z
∂ p = 0 Thus ρ, p and p must be independent of z.
z z r T
The system of inhomogeneous dark radiation and an effective cosmological constant is defined
by a conformal bulk matter with the following equations of state
Λ Λ
ρ = −p = ρ + and p +ηρ = (η−1), (29)
r DR κ2 T κ2
5 5
where η characterizes the dark radiation model and Λ is a bulk quantity and mimics a 4D cosmo-
logical constant on the brane. The components of the Einstein tensor can be thus evinced:
Gr = Gt = −κ2ρ −Λ, Gθ = Gφ = −κ2ηρ −Λ, (30)
r t 5 DR θ φ 5 DR
Gz = −κ2ρ (1+η)−2Λ. (31)
z 5 DR